2012/12/11 by Chleboun, Paul, Faggionato, Alessandra, Martinelli, Fabio
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.1212.2399
We consider the non-equilibrium dynamics of the East model, a linear chain of 0-1 spins evolving under a simple Glauber dynamics in the presence of a kinetic constraint which forbids flips of those spins whose left neighbor is 1. We focus on the glassy effects caused by the kinetic constraint as q\downarrow 0, where q is the equilibrium density of the 0's. In the physical literature this limit is equivalent to the zero temperature limit. We first prove that, for any given L=O(1/q), the divergence as q\downarrow 0 of three basic characteristic time scales of the East process of length L is the same. Then we examine the problem of dynamic heterogeneity, i.e. non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium, one of the central aspects of glassy dynamics. For any mesoscopic length scale L=O(q-γ), γ<1, we show that the characteristic time scale of two East processes of length L and λL respectively are indeed separated by a factor q-a, a=a(γ)>0, provided that λ≥ 2 is large enough (independent of q, λ=2 for γ<1/2). In particular, the evolution of mesoscopic domains, i.e. maximal blocks of the form 111..10, occurs on a time scale which depends sharply on the size of the domain, a clear signature of dynamic heterogeneity. A key result for this part is a very precise computation of the relaxation time of the chain as a function of (q,L), well beyond the current knowledge, which uses induction on length scales on one hand and a novel algorithmic lower bound on the other. Finally we show that no form of time scale separation occurs for γ=1, i.e. at the equilibrium scale L=1/q, contrary to what was assumed in the physical literature based on numerical simulations.